Sandwich-type theorems for a class of integral operators

Let H(U) be the space of all analytic functions in the unit disk U. For a given function h epsilon A we define the integral operator I-h;beta : K -> H(U), with K subset of H (U), by Ih;beta[f](z) = [beta integral(z)(0)f(beta)(t)h-1(t)h'(t)d t](1/beta), where beta epsilon C and all powers are the principal ones. We will determine sufficient conditions on g(1), g(2) and beta such that [zh'(z)/h(z)](1/beta) g(1)(z) < [zh'(z)/h(z)](1/beta) f(z) < [zh'(z)/h(z)](1/beta) g(2)(z) implies I-h;beta[g(1)](z) < I-h;beta[f](z) < Ih;beta[g(2)](z), where the symbol "<" stands for subordination. We will call such a kind of result a sandwich-type theorem. In addition, I-h;beta[g(1)] will be the largest function and I-h;beta[g1] the smallest function so that the left-hand side, respectively the right-hand side of the above implication hold, for all f functions satisfying the differential subordination, respectively the differential superordination of the assumption. We will give some particular cases of the main result obtained for appropriate choices of the h, that also generalize classic results of the theory of differential subordination and superordination. The concept of differential superordination was introduced by S. S. Miller and P. T. Mocanu in [5] like a dual problem of differential subordination [4].